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arXiv · 2112.00816

Maximum Likelihood Estimation for Brownian Motion Tree Models Based on One Sample

Michael Truell·Jan-Christian Hütter·Chandler Squires·Piotr Zwiernik·Caroline Uhler
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Abstract

We study the problem of maximum likelihood estimation given one data sample ($n=1$) over Brownian Motion Tree Models (BMTMs), a class of Gaussian models on trees. BMTMs are often used as a null model in phylogenetics, where the one-sample regime is common. Specifically, we show that, almost surely, the one-sample BMTM maximum likelihood estimator (MLE) exists, is unique, and corresponds to a fully observed tree. Moreover, we provide a polynomial time algorithm for its exact computation. We also consider the MLE over all possible BMTM tree structures in the one-sample case and show that it exists almost surely, that it coincides with the MLE over diagonally dominant M-matrices, and that it admits a unique closed-form solution that corresponds to a path graph. Finally, we explore statistical properties of the one-sample BMTM MLE through numerical experiments.

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Michael Truell, Jan-Christian Hütter, Chandler Squires, Piotr Zwiernik, Caroline Uhler. 2021-12-01. Maximum Likelihood Estimation for Brownian Motion Tree Models Based on One Sample. https://arxiv.org/abs/2112.00816

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